Graphs and the Derivative. Applications of the Derivative including Extrema, Implicit Differentiation, Related Rates & Linear Approximation

Paper details:

The rules my teacher gave me are: Question 1: – Follow the correct order of the steps. Don’t skip any step. – Label the axes, asymptotes and all key points on the graph. You don’t need to use graph paper. Question 3: – Remember to define your variables and identify the objective function. Question 4: – Write the correct notation for derivative, eg. d/dx (3x + 2y) for derivative with respect to x of (3x+2y). The rules my teacher refers to in Question 1 are: To sketch the graph of a function ƒ: 1. Consider the domain of the function, and note any restrictions. (That is, avoid dividing by 0, taking a square root, or any even root, of a negative number, or taking the logarithm of 0 or a negative number.) 2. Find the y-intercept (if it exists) by substituting x = 0 into ƒ(x). Find any x-intercepts by solving ƒ(x) = 0 if this is not too difficult. 3. (a) If ƒ is a rational function, find any vertical asymptotes by investigating where the denominator is 0, and find any horizontal asymptotes by finding the limits as x—> ∞ and x —> -∞. (b) If ƒ is an exponential function, find any horizontal asymptotes; if ƒ is a logarithmic function, find any vertical asymptotes. 4. Investigate symmetry. If ƒ(-x) = ƒ(x), the function is even, so the graph is symmetric about the y-axis. If ƒ(-x) = -ƒ(x) the function is odd, so the graph is symmetric about the origin. 5. Find ƒ′(x). Locate any critical points by solving the equation ƒ′(x) = 0 and determining where ƒ′(x) does not exist, but ƒ1×2 does. Find any relative extrema and determine where ƒ is increasing or decreasing. 6. Find ƒ′′(x). Locate potential inflection points by solving the equation ƒ′′(x) = 0 and determining where ƒ′′(x) does not exist. Determine where ƒ is concave upward or concave downward. 7. Plot the intercepts, the critical points, the inflection points, the asymptotes, and other points as needed. Take advantage of any symmetry found in Step 4. 8. Connect the points with a smooth curve using the correct concavity, being careful not to connect points wheagnre the function is not defined.